Solve the simultaneous equations:
y = x2 - x - 1
y = x + 2
y = x2 - x - 1
y = x + 2
Worked Solution & Answer
Equating y: x2 - 2x - 3 = 0 ⇒ (x - 3)(x + 1) = 0 B1
Answer: (3, 5) and (-1, 1)
Quad. sim. equations — Notes & Worked Examples Show / Hide
Explanation:
To solve simultaneous equations where one equation is linear and the other is quadratic (such as a circle x2 + y2 = r2 or parabola), always use substitution: rearrange the linear equation to make x or y the subject, then substitute this expression into the quadratic equation.
Expand the brackets carefully, rearrange into standard quadratic form ax2 + bx + c = 0, and solve by factorising or the quadratic formula. Finally, substitute each root back into the linear equation and pair up your (x, y) coordinates clearly.
Example 1: Solve y = x2 − 3x + 4 and y = 2x − 2.
Solution:
Equate y: x2 − 3x + 4 = 2x − 2
Rearrange to 0: x2 − 5x + 6 = 0 ⇒ (x − 2)(x − 3) = 0
So x = 2 or x = 3
In y = 2x − 2: when x = 2, y = 2; when x = 3, y = 4
Answer: (2, 2) and (3, 4)
Example 2: Solve x2 + y2 = 25 and y − 2x = 5.
Solution:
Substitute y = 2x + 5 into x2 + y2 = 25:
x2 + (2x + 5)2 = 25 ⇒ 5x2 + 20x = 0
5x(x + 4) = 0 ⇒ x = 0 (y = 5) or x = −4 (y = −3)
Answer: (0, 5) and (−4, −3)
Solve the simultaneous equations:
y = x2 + 2x
y = x + 2
y = x2 + 2x
y = x + 2
Worked Solution & Answer
Equating y: x2 + x - 2 = 0 ⇒ (x - 1)(x + 2) = 0 B1
Answer: (1, 3) and (-2, 0)
The diagram shows the graph of x2 + y2 = 30.25
Use the graph to find estimates for the solutions of the simultaneous equations
x2 + y2 = 30.25
y - 2x = 1
x2 + y2 = 30.25
y - 2x = 1
Worked Solution & Answer
Plot line y = 2x + 1 through (0, 1) and (2, 5) M2
Intersections with circle are at (2.1, 5.1) and (–2.9, –4.7). A1
Intersections with circle are at (2.1, 5.1) and (–2.9, –4.7). A1
Answer: x = 2.1, y = 5.1 and x = -2.9, y = -4.7
Solve the simultaneous equations
y2 = 3x2 + 4
y + 2x = 7
Give your solutions correct to 3 significant figures.
y2 = 3x2 + 4
y + 2x = 7
Give your solutions correct to 3 significant figures.
Worked Solution & Answer
y = 7 – 2x M1
⇒ (7 – 2x)² = 3x² + 4 M1
⇒ 49 – 28x + 4x² = 3x² + 4
⇒ x² – 28x + 45 = 0 M1
⇒ x = (28 ± √(784 – 180))/2 = 14 ± √151
⇒ x = 26.288 (y = –45.576) or x = 1.712 (y = 3.576 ≈ 3.58) A1
⇒ (7 – 2x)² = 3x² + 4 M1
⇒ 49 – 28x + 4x² = 3x² + 4
⇒ x² – 28x + 45 = 0 M1
⇒ x = (28 ± √(784 – 180))/2 = 14 ± √151
⇒ x = 26.288 (y = –45.576) or x = 1.712 (y = 3.576 ≈ 3.58) A1
Answer: x = 26.3, y = -45.6 and x = 1.71, y = 3.58
Solve algebraically the simultaneous equations
x2 - 4y2 = 9
3x + 4y = 7
x2 - 4y2 = 9
3x + 4y = 7
Worked Solution & Answer
4y = 7 – 3x M1
⇒ y = (7 – 3x)/4. Substitute into x² – 4y² = 9: x² – (7 – 3x)² / 4 = 9 M1
⇒ 4x² – (49 – 42x + 9x²) = 36 M1
⇒ 5x² – 42x + 85 = 0 M1
⇒ (5x – 17)(x – 5) = 0
⇒ x = 5, y = –2 or x = 3.4, y = –0.8 A1
⇒ y = (7 – 3x)/4. Substitute into x² – 4y² = 9: x² – (7 – 3x)² / 4 = 9 M1
⇒ 4x² – (49 – 42x + 9x²) = 36 M1
⇒ 5x² – 42x + 85 = 0 M1
⇒ (5x – 17)(x – 5) = 0
⇒ x = 5, y = –2 or x = 3.4, y = –0.8 A1
Answer: x = 5, y = -2 and x = 325, y = -45
Solve algebraically the simultaneous equations
\begin{aligned}3x2 + 2y2 &= 44 \\ 3x + y &= 2\end{aligned}
\begin{aligned}3x2 + 2y2 &= 44 \\ 3x + y &= 2\end{aligned}
Worked Solution & Answer
From linear equation: y = 2 – 3x M1
Substitute into quadratic: 3x² + 2(2 – 3x)² = 44
3x² + 2(4 – 12x + 9x²) = 44 M1
3x² + 8 – 24x + 18x² = 44
21x² – 24x – 36 = 0 M1
Divide by 3: 7x² – 8x – 12 = 0
Factorise: (7x + 6)(x – 2) = 0 ⇒ x = 2 or x = –6/7. M1
When x = 2: y = 2 – 3(2) = –4.
When x = –6/7: y = 2 – 3(–6/7) = 2 + 18/7 = 32/7. A1
Substitute into quadratic: 3x² + 2(2 – 3x)² = 44
3x² + 2(4 – 12x + 9x²) = 44 M1
3x² + 8 – 24x + 18x² = 44
21x² – 24x – 36 = 0 M1
Divide by 3: 7x² – 8x – 12 = 0
Factorise: (7x + 6)(x – 2) = 0 ⇒ x = 2 or x = –6/7. M1
When x = 2: y = 2 – 3(2) = –4.
When x = –6/7: y = 2 – 3(–6/7) = 2 + 18/7 = 32/7. A1
Answer: x = 2, y = -4 and x = -67, y = 327
L is the straight line with equation y = 2x - 5
C is a graph with equation y2 = 6x2 - 25x - 8
Using algebra, find the coordinates of the points of intersection of L and C.
You must show all your working.
C is a graph with equation y2 = 6x2 - 25x - 8
Using algebra, find the coordinates of the points of intersection of L and C.
You must show all your working.
Worked Solution & Answer
(2x – 5)² = 6x² – 25x – 8 ⇒ 4x² – 20x + 25 = 6x² – 25x – 8 ⇒ 2x² – 5x – 33 = 0 ⇒ (2x – 11)(x + 3) = 0 ⇒ x = 5.5 or x = –3. When x = –3, y = 2(–3) – 5 = –11 M4
when x = 5.5, y = 2(5.5) – 5 = 6 A1
when x = 5.5, y = 2(5.5) – 5 = 6 A1
Answer: (–3, –11) and (5.5, 6)
Solve the simultaneous equations
\begin{aligned}x2 - 9y - x &= 2y2 - 12 \\ x + 2y - 1 &= 0\end{aligned}
\begin{aligned}x2 - 9y - x &= 2y2 - 12 \\ x + 2y - 1 &= 0\end{aligned}
Show clear algebraic working.
Worked Solution & Answer
From linear equation: x = 1 – 2y M4
Substitute into quadratic: (1 – 2y)² – 9y – (1 – 2y) = 2y² – 12 => 1 – 4y + 4y² – 9y – 1 + 2y = 2y² – 12 => 2y² – 11y + 12 = 0 => (2y – 3)(y – 4) = 0 => y = 3/2 or y = 4. When y = 3/2, x = 1 – 3 = –2. When y = 4, x = 1 – 8 = –7. A1
Substitute into quadratic: (1 – 2y)² – 9y – (1 – 2y) = 2y² – 12 => 1 – 4y + 4y² – 9y – 1 + 2y = 2y² – 12 => 2y² – 11y + 12 = 0 => (2y – 3)(y – 4) = 0 => y = 3/2 or y = 4. When y = 3/2, x = 1 – 3 = –2. When y = 4, x = 1 – 8 = –7. A1
Answer: x = -2, y = 1.5 and x = -7, y = 4
(a) On the grid, draw the graph of x² + y² = 169 (2) (b) Use your graph to find estimates for the solutions of x² + y² = 169 and 2y = 3x (3)
(a) On the grid, draw the graph of x2 + y2 = 169 [2 marks]
(b) Use your graph to find estimates for the solutions of the simultaneous equations
x2 + y2 = 169
2y = 3x [3 marks]
x2 + y2 = 169
2y = 3x [3 marks]
Worked Solution & Answer
(a)
√169 = 13, draw circle centre (0, 0) radius 13 M1 A1
Answer (a): Circle drawn with centre (0, 0) and radius 13
(b)
Draw line y = 1.5x M2
intersections at (7.2, 10.8) and (–7.2, –10.8) A1
intersections at (7.2, 10.8) and (–7.2, –10.8) A1
Answer (b): x = 7.2, y = 10.8 and x = -7.2, y = -10.8
(a) Draw graph of x² + y² = 12.25 (2) (b) Hence find estimates for solutions of x² + y² = 12.25 and 2x + y = 1 (3)
(a) On the grid, draw the graph of x2 + y2 = 12.25 [2 marks]
(b) Hence find estimates for the solutions of the simultaneous equations
x2 + y2 = 12.25
2x + y = 1 [3 marks]
x2 + y2 = 12.25
2x + y = 1 [3 marks]
Worked Solution & Answer
(a)
r = √12.25 = 3.5 M1 A1
Answer (a): Circle drawn with centre (0, 0) and radius 3.5
(b)
Draw line y = 1 – 2x M2
intersections with circle are at (2.0, –2.9) and (–1.2, 3.3) A1
intersections with circle are at (2.0, –2.9) and (–1.2, 3.3) A1
Answer (b): x = 2.0, y = -2.9 and x = -1.2, y = 3.3